Decidability and complexity of fibred logics without shared connectives

Abstract

Fibring is a powerful mechanism for combining logics, and an essential tool for designing and understanding complex logical systems. Abstract results about the semantics and proof-theory of fibred logics have been extensively developed, including general preservation results for metalogical properties like soundness and (sufficient conditions for) completeness. Decidability, however, a key ingredient for the automated support of the fibred logic, has not deserved similar attention. In this paper, we extend the preliminary results obtained in [28] regarding theoremhood, and address the problem of deciding the consequence relation of fibred logics, assuming that the logics do not share connectives. Namely, under this assumption, we provide a thorough characterization of the mixed patterns of reasoning that can occur in the fibred logic, and use it to obtain the first general decidability preservation result for fibring. The complexity of the decision procedure we obtain is also analyzed.

Merging Fragments of Classical Logic

“…Rautenberg's general method for axiomatizing fragments of classical logic [9], which explores the structure of Post's lattice [8,5], further confirms the intuition about the essential role of interaction axioms/rules, that one may have drawn from any experience with axiomatizations of classical logic. Additionally, such expectation is consistent with a careful analysis of the characterization of the complexity of different fragments of classical logic and their associated satisfiability problems [10,12], namely in the light of recent results on the decidability and complexity of fibred logics [6]. The question we wish to give a definitive answer to, here, is precisely this: is it possible to recover classical logic by fibring two disjoint fragments of it?…”

“…In a natural conservative extension, where the syntax of a logic is extended with new connectives but no further inference power is added, it is clear that formulas headed by the newly added connectives are treated as monoliths. Hence, the following result from [6] applies:…”

“…Note that the roles of a and b may be exchanged in the above theorem, given that the fibring operation is obviously commutative, so we might talk accordingly of a corresponding condition (Z b ), in case it turns out to be more convenient. The original formulation of this result in [6] was based on a slightly more sophisticated notion of saturation, which reduces to the above one in particular when the logics involved in the combination are characterizable by means of a truth-functional semantics (i.e., a matrix semantics involving a single logical matrix), as it is indeed the case for all sublogics of classical logic. The following is the first useful new result of this paper, establishing that conservativity is preserved by disjoint fibring, here proved for the (slightly simpler) case where each logic is characterized by a single logical matrix.…”

“…Some unexpected situations do pop up, like the fact that B ↔ • B ⊥ = B ↔ ⊥ , or the fact that B → • B ⊤ and B ∨+ • B ⊤ both yield full classical logic. The latter two combinations are particularly enlightening, given that according to [6] the complexity of disjoint fibring is only polynomially worse than the complexity of the component logics, and we know from [1] that the decision problems for B → or B ∨+ are both co-NP-complete, as in full classical logic. As a matter of fact, some of the results we obtained may alternatively be established as consequences of the complexity result in [6] together with the conjecture that P = NP.…”

“…The latter two combinations are particularly enlightening, given that according to [6] the complexity of disjoint fibring is only polynomially worse than the complexity of the component logics, and we know from [1] that the decision problems for B → or B ∨+ are both co-NP-complete, as in full classical logic. As a matter of fact, some of the results we obtained may alternatively be established as consequences of the complexity result in [6] together with the conjecture that P = NP. In fact, for disjoint sets of Boolean connectives conn 1 and conn 2 such that conn 1 ∪ conn 2 is functionally complete, if the decision problems for B conn 1 and for B conn 2 are both in P then clearly B conn 1 • B conn 2 = B conn 1 ∪conn2 .…”